The juggler throws n balls per second.

$$\therefore$$ Interval between balls = $${1 \over n}$$ seconds

At maximum height velocity of first ball $$v = 0$$

Juggler throws all balls with same initial velocity $$ = u$$

For 1st ball,

$${u_1} = u$$, $${v_1} = 0$$, $$a = - g$$, $$S = {H_{\max }}$$

Using formula,

$${v^2} = {u^2} + 2as$$

$$0 = {u^2} - 2g\,{H_{\max }}$$

$$ \Rightarrow {H_{\max }} = {{{u^2}} \over {2g}}$$ ...... (1)

And using formula,

$$v = u + at$$

$$ \Rightarrow 0 = u - gt$$

$$ \Rightarrow t = {u \over g}$$

$$\therefore$$ Time taken by ball 1 to reach maximum height $$({H_{\max }}) = {u \over g}$$

According to the question,

$$t = {1 \over n}$$

$$ \Rightarrow {u \over g} = {1 \over n}$$

$$ \Rightarrow u = {g \over n}$$ ....... (2)

Putting value of $$u$$ in equation (1), we get

$${H_{\max }} = {{{{\left( {{g \over n}} \right)}^2}} \over {2g}}$$

$$ = {{{g^2}} \over {2{n^2}g}}$$

$$ = {g \over {2{n^2}}}$$